IFSA/NAFIPS 2019 2019: Fuzzy Techniques: Theory and Applications pp 246-257

# Some Notes on the Addition of Interactive Fuzzy Numbers

• Estevão Esmi
• Laécio Carvalho de Barros
• Vinícius Francisco Wasques
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1000)

## Abstract

This paper investigates some fundamental questions involving additions of interactive fuzzy numbers. The notion of interactivity between two fuzzy numbers, say A and B, is described by a joint possibility distribution J. One can define a fuzzy number $$A +_J B$$ (or $$A -_J B$$), called J-interactive sum (or difference) of A and B, in terms of the sup-J extension principle of the addition (or difference) operator of the real numbers. In this article we address the following three questions: (1) Given fuzzy numbers B and C, is there a fuzzy number X and a joint possibility distribution J of X and B such that $$X +_J B = C$$? (2) Given fuzzy numbers AB,  and C, is there a joint possibility distribution J of A and B such that $$A +_J B = C$$? (3) Given a joint possibility distribution J of fuzzy numbers A and B, is there a joint possibility distribution N of $$(A +_J B)$$ and B such that $$(A +_J B) -_N B = A$$? It is worth noting that these questions are trivially answered in the case where the fuzzy numbers A, B and C are real numbers, since the fuzzy arithmetic $$+_J$$ and $$-_N$$ are extension of the classical arithmetic for real numbers.

## Notes

### Acknowledgements

This work was partially supported by FAPESP under grant no. 2016/26040-7 and CNPq under grants no. 306546/2017-5 and 142414/2017-4.

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## Authors and Affiliations

• Estevão Esmi
• 1
Email author
• Laécio Carvalho de Barros
• 1
• Vinícius Francisco Wasques
• 1
1. 1.Institute of Mathematics, Statistics and Scientific ComputingUniversity of CampinasCampinasBrazil